We reexamine light cone quantisation of scalar field theories with a spontaneously broken internal symmetry. To this end we use the Dirac-Bergmann algorithm for quantisation of constrained systems. Carefully taking into account the zero modes with light cone momentum k+ = 0, we obtain secondary constraints which determine the (tree level) vacuum structure of the theories. In 1 + 1 dimensions spontaneous symmetry breaking shows up in terms of a direct sum decomposition of the Hilbert space corresponding to the discrete phases of the theory. In 3 + 1 dimensions, when a continuous symmetry is spontaneously broken, one finds additional first class constraints corresponding to the broken group generators.